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Implement classical MDS #31322

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Implement classical MDS #31322

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@dkobak dkobak commented May 6, 2025

Fixes #15272. Supersedes #22330.

This PR implements classical MDS, also known as principal coordinates analysis (PCoA) or Torgerson's scaling, see https://en.wikipedia.org/wiki/Multidimensional_scaling#Classical_multidimensional_scaling. As discussed in #22330, it is implemented as new class ClassicalMDS.

Simple demonstration:

import pylab as plt
import numpy as np

from sklearn.datasets import load_iris
from sklearn.manifold import ClassicalMDS, MDS
from sklearn.decomposition import PCA

X, y = load_iris(return_X_y=True)

Z1 = PCA(n_components=2).fit_transform(X)
Z2 = ClassicalMDS(n_components=2, dissimilarity="euclidean").fit_transform(X)
Z3 = ClassicalMDS(n_components=2, dissimilarity="cosine").fit_transform(X)
Z4 = ClassicalMDS(n_components=2, dissimilarity="manhattan").fit_transform(X)

fig, axs = plt.subplots(nrows=2, ncols=2, figsize=(6, 6), layout="constrained")

axs.flat[0].scatter(Z1[:,0], Z1[:,1], c=y)
axs.flat[0].set_title("PCA")

axs.flat[1].scatter(Z2[:,0], Z2[:,1], c=y)
axs.flat[1].set_title("Classical MDS, Euclidean dist.")

axs.flat[2].scatter(-Z3[:,0], Z3[:,1], c=y)
axs.flat[2].set_title("Classical MDS, cosine dist.")

axs.flat[3].scatter(Z4[:,0], Z4[:,1], c=y)
axs.flat[3].set_title("Classical MDS, Manhattan dist.")

cmds

For consistency, this PR also adds support for non-Euclidean metrics to the MDS class.

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dkobak commented May 7, 2025 •
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Note 1: I assume this PR won't make it in time for 1.7, which is why I wrote 1.8 in versionchanged. But I am of course happy if it does get merged into 1.7.

Note 2: I think it would make sense to use ClassicalMDS() as the default initialization for MDS(), making this:

MDS().fit(X, init=ClassicalMDS().fit(X))

the default behavior of

MDS().fit(X)

Currently I have not implemented it because I wasn't sure about the best API. MDS() constructor could get a new init={"random", "classical_mds"} parameter, but then fit(init=Z) would need to be able to override that... For comparison, in TSNE() class, init is the parameter of the constructor, not of the fit() function. We could also do that here and deprecate init parameter of the fit()... Not sure it's the most sensible option though. But I do think that TSNE() and MDS() should have the same API with regard to initialization.

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sklearn MDS vs skbio PCoA
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